161
r r H E SEDIhlENTATION VELOCITY EXPEHIMENT
For series A, plots of log [ E ] ,us. time resulted jn a family of straight lines intersecting a t a point corresponding to 33 min. and 0.206 mequiv./ml. for all concentrations up to 3 M . For series B, the common point of intersection for lines obtained with aqueous phases less concentrated than 3 M NaOH was 34 min. and 0.285 mequiv./ml. In both series, the straight lines for the two most concentrated basic solutions used with RDO did not intersect a t the point of intersection of all the other lines. The ratio of the epoxide concentration a t the common point of series A to that of series B (0.206/ 0.285) is 0.723. Molar ratios of BDO in the organic layer
of series A to that in series B a t equilibria conditions before finite amounts of epoxide have hydrolyzed can be calculated by use of the distribution coefficient, 0.52, obtained for the distribution of the diepoxide between CCl, arid water phases. Use of K = 0.52 resulted in a molar ratio of 0.73, in agreement with that obtained experimentally, thus indicating that distribution equilibrium is established relatively fast compared to the rate a t which the diepoxide hydrolyzes in the aqueous phases for those aqueous phases less concentrated than 3 M KaOH.
The Sedimentation Velocity Experiment and the Determination of Molecular Weight Distributions
by James E. Blair and J. W. Williams Laboratory o j Physical Chemistry, University o j Wisconsin, Madiaon, Wisconsin
(Received August 21, 1903)
Observations of boundary spreading in sedimentation velocity experiments for the system polystyrene-cyclohexane a t the Flory temperature have been utilized largely to demonstrate a procedure whereby solute molecular weight distributions may be obtained. The matter of using the movement of the position of the maximum height of the boundary gradient curve instead of the boundary location as measured from the second moment for the determination of the pressure dependent parameter in sedimentation is considered In some detail. In addition, several other items such as the determination of the zero time correction are discussed.
Introduction Along with its other attributes, the ultracentrifuge is an instrument by which there may be achieved a physical fractionation of solute molecules of different molecular weights along the radial axis in the rotating cell. Indeed, the possibility so provided of making a molecular weight analysis of a mixture is one of the great advantages of the ultracentrifugal techniques. One may determine the several more common average
molecular weights; in addition and now being considered, procedures are made available by which a complete molecular weight distribution curve may be constructed. In principle, the necessary data may be provided either a t sedimentation equilibrium or during sedimentation transport. The methods which involve the sedimentation equilibrium experiment have been studied by Rindel (1)
€1. Rinde, Dissertation, Vppsala, 1928.
Volume 68.iVumber 1
January, 1.964
162
JAMES E. BLAIRA N D J. Mi. WILLIAMS
and by Wales and Williams and their associates.2 Using polyst.yrene as solute it could be shown2c that the molecular weight distribution curve obtained by this route compared well with that constructed from the record of fractional precipitations. However, there are reasons to believe that the sedimentation transport experiment is the more advantageous from the point of view of resolution and we present here the record of an analysis in which it is used and by which the molecular weight distribution for a typical linear organic high polymer was achieved. It is based iipoti two ideas, that of a boundary spreading analysis to o h i n a distribution of sedimentation coefficient, s, and that of a unique continuous correspondance between sedimentation coefficient and molecular weight for the transformation to a distribution of Obscrvations of the behavior molecular weight, A[. and propert,ies of the syst,em polystyrene--cyclohexane a t the Id‘Iory tempcrature were used. As far. as we are aware, the original outline for this mode of analysis was described by one of the present authors.” RIore recently, it has been the subject of substantial development from several points of view including pressure corrections, t,he use of systems a t the E’lory temperature to avoid concentration dependence effects, computational methods, ~ t c . ~ - 8
Theory The equations descriptive of the transformation of the boundary gradient curve of the sedimentation transport experiment into a distribution of sedimentation coef€icients now have appeared in a number of places. Following tho presentation of the original and fundamental equation for the distribution by Bridgman9 there were developed’n-12 theory and procedures for taking into account the effects of diffuion on the measurement of heterogeneity. However, it soon appeared that, in general, concentration-dependence effects must be also considered in finding the true distribution function by means of an extrapolation for which there is a t present no really gsatisfactory thcorctical ba~is.13,’~Ail excellent survey of the present st’ntusof the intcrprctation of boundary spreadirig in sedimentation velocity experiments is t o be found in the rccerit Fujita monograph. l6 If for the present the effects of diffusion and concentration dependence are ignored, the distribution of sedimentation coeficient, s, is given by dc/ds = (dc/dr) (dy,’ds), except for the radial dilution factor which causes this quantity to diminish with time. To compensate, we write 1 dc dr r b 2 ~ Z (t to)r dc/dr - = s*(s) = .(1) cI, dr ds cor2 I
The Journal of Physical Chemistry
where c, represents the decreased concentration of the solution in the “plateau” region, and co is the original concentration of the solution. The asterisk indicates that the distribution has not been corrected for diffusion and concentration drpendence; the other symbols have their usual significance. In working with solutions of organic high polymers a t the Plory temperature, it is entirely proper to neglect concentration dependence effects on boundary spreading. Indeed, it is this fact which is probably the greatest single factor in contributing to the succcss of the analysis for polydispersity. Also, in systems of this kind diffusion effects are relatively unimportant. However, the organic solvents which now must be used are relatively compressible substances as compared to water, and an additional complication is introduced. It is the effect, of hydrostatic pressure on the viscosity and the density of this solvent, and on the partial specific volume of the solute as well. The net result is a variation of the sedimentation coefficient with distance in the cell, one which is accounted for by a single prcssure dependent parameter. l6 F ~ j i t a ’has ~ developed the equation which relates movement of the boundary position, T , with time, t , for the case of isothermal sedimentation in a system in which the sedimentation coefficient depends on both hydrostatic pressure and concentration. He writes _ _In r/ro = soco (1 K ;J2 -1 (2) W2(t
- to)
+
[(
]>
where
~~
~
(a) >I. Wales, M.h l . Bender, J . R. Williams, and R. H. Ewart, J . Chem. Phys., 14, 353 (1946); (b) M. Wales, J . Phys. Colloid Chem., 52, 235 (1948); (c) M. Wales, J. W. Williams, J. 0. Thompson, and It. H. Ewart, ihid.,52,983 (1948); (d) M.Wales, F. T. Adler, and K. E. Van Holde. ibid., 55, 145 (1951). (3) J. W. Williams, J . Polymer Sci., 12, 351 (1954). (4) A. 17. V. Uriksson, Acta Chem. Scand., 10, 360 (1956). (5) H, W. ,McCormick, J . Polymer Sci., A l , 103 (1963). (6) I. H. Billick, ibid,,62, 167 (1962). (7) M.Wales and 8.J. Rehfeld, ibid., 6 2 , 179 (1962). ( 8 ) G . Meyerhoff, in “Ultracentrifugal Analysis in Theory and E x periment,” Acudemic Press, New York and London, 1963. (9) W. B. Bridgman, J . A m . Chmn. Soc., 64, 2349 (1942). (10) R. L. Raldwin and J. W. Williams, ibid.. 72, 4325 (1950). (11) J. W. Williams, I t . L. Baldwin, W. M. Saunders, and P. 0. Squire, ibid.. 74, 1642 (1952). (12) L. J. Gosting, ihid., 74, 1548 (1952). (13) J. W. Williams and W. M. Saunders, J . Phya. Chem., 58, 854 (2)
(1954).
It. L. Baldwin, J . A m . Chem. Soe., 76, 402 (1964). H. Fujita, “Mathematical Theory of Sedimentation Analysis,” Academic. Press. Inc., New York, N. Y . , 1962. (16) J. 0 t h and V. Desreux, Bull. a m . chim. Belges, 63, 133 (1954).
(14) (15)
T H E SE1)IhPElrjTATION J'ELOCITY
EXPERIMENT
163
Actually, this equation has been derived to describe the behavior of a monodisperse solute system with the effects of diffusion being neglected. By numerical computations we have justified its later use in connection with polydisperse systems, using the boundary position ~b as the proper value of r. Also, in eq. 2, T~ is the radial distance from the axis of rotation to the meniscus, t is the time as observed, lo is the zero time concentration, w is the angular velocity, and socais the value of the sedimentation coefficient a t 1 atm. pressure (superscript zero) and a t the initial concentration of the solution. The concentration dependence parameter k is defined by the expression
(3) whcrc soo is the value of so a t infinite dilution. The quantity m, the pressure dependence parameter, takes the form'6
m = 1/*w2p,,"ro2 (4) where p00 is the density of the solvent and ~.ris a constant characteristic of the solute-solvent system. For the system polystyrene in cyclohexane at 34.2' (the Flory or &temperature) the concentration dependence of s is small compared to that of the pressure dependence and to the limit of precision in determining the constant K
K = -m/2 (5) From the usual definition of s (eq. 2 with K = 0), one may obtain for use in eq. 1 dr - = royt - to) (6) ds Now, if the effect of hydrostatic pressure (but not concentration dependence) is included, eq. 6 is modified to read
Thus
For the usual values of the parameters which occur in sedimentation velocity experiments, eq. 7s can be also written as dr -ds
=
ruyt - lo)[{l
-2);([;
-
-1
srm d r To2 ds
or
-
To the degree of approximation that terms in
[(-:)'-
1 1 and in higher powers of this quantity can
be neglected -.r2 To2
In --T r0
=-[ (k)2 1 2
-
11
(8)
With the introduction of eq. 8 into eq. 7 we find
Finally, the combination of eq. 9 and 1 gives
(10)
This equation is equivalent to forms already arrived a t by Billick6 and by Wales and Rehfeld,' since in the case of pressure dependent sedimentation the radial dilution law takes thc form
The analysis we use presupposes in addition a knowledge of the relation between limiting sedimentation coefficient and molecular weight so that there is made available a method of transformation of g(so) to j ( M ) , the molecular weight distribution. The relation s =
-
I]}
KMa
(12)
may be used. Then, if we consider continuous distributions it is possible to write
(7)
g(s")ds
=
f(M)dM
(13)
Making the substitution
;{(;)'
r w y t - to)[l - l}] dr - ---____ -_ mr2 In r/ro ds 1+
I
To2
[l -2):({;
- l}]
For the system polystyrene-cyclohexane a t 34.2', tho Flory temperature, the constants a and K have been Volume 68, Sumher 1
Januarg. 1864
JAMES E. BLAIRANI) J. W. WILLIAMS
164
determined by both lIcCormick17 and by Cantow.@ These data are given, in order, by the equations s
=
s
=
x (1.35 x
(1.69
10-2)
1&!0.4*
10-2) MO.5’
(124 (12b)
The equivalent sphere model for the friction coefficient of flexible chain polymers predicts a value of cy = 0.50 for the poor solvent.
Experimental Two samples of polystyrene were used in the experiments. One of them, an anionically polymerized material of designation S-103, was furnished to us through the courtesy of Dr. H. W. McCormick of the Dow Chemical Co. The other specimen, l Y F , was provided by Prof. J. D. Ferry of this Department. It was originally distributed by Dr. It. F. Boyer, also of Dow Chemical Co., as a preparation having substantially a most probable molecular weight distribution. Both polystyrenes have been the subject of several prior average molecular weight determinations. Polymer solutions in cyclohexane a t 34.2’, the Flory temperature, were studied during sedimentation transport in a Spinco Model E ultracentrifuge. I n all experiments double sector cells were used, with solution on one side and solvent on the other, and with both sides being filled to approximately the same level. Four rotational speeds were used : 59,780, 56,100, 47,660, and 39,460 r.p.m. A schlieren optical system was employed to record the redistribution of the components during the sedimentation process. In the earlier experiments great care was taken to remove water from the cyclohexane. The solvent was distilled from lithium aluminum hydride and the polymer solutions were prepared in a drybox. However, when data taken in this way were compared with those when these precautions were not observed, no consistent or appreciable diff erences were found. So, these precautions were eventually omitted. All solutions were prepared by weighlng the polymer samplo into a dO-ml. long-neck, glass-stoppered flask. The solvent was introduced into the, flask and the weight of the solvent was determined by a second weighing. The contents of the flask were then frozen in a Dry Ice bath and the neck of the flask w, sealed off. The flask was then placed in an oven at 4 . ~ ’with gentle stirring for at least 1 day to ensure complete solution. After restoration to near 0-temperature, the flask was opened and the solution was removed by hypodermic syringe for insertion into the prewarmed cell. In computing the concentration of a solution, the density of the cyclohexane at 34.2’ was taken to be The Journal of Physical Chemietry
0.7656 g./ml. and the partial specific volume of the dissolved polystyrene was assumed to be 0.95 cc./g. The times, t , were measured from the moment the acceleration of the rotor commenced. The acceleration was maintained constant during the speedup period and the times a t which the rotor reached two-thirds of its final speed were recorded for comparison with the zero-time correction, 6, as ultimately computed. Several methods for the evaluation of this quantity were considered, it having been found that the value of the pressure and concentration dependence parameter K , eq. 2 , is rather sensitive to the value of to applied. In this work, the value of to selected was the one which produced the “best fit” of the data to eq. 2; ie., the to which when inserted produced the minimum standard deviation, u, defined by
(15) with the residual, R, being defined as
T
In --
(15a)
TO
The position, r = Tb, is the point on the moving boundary gradient curve which has the same velocity as does the corresponding solute molecule ahead of the boundary. The values of rt, were determined by numerical integration of the points along the boundary gradient curve. These, along with the times t which correspond, were fitted to eq. 2 by the method of least squares to determine the quantities socoand K . The values of these quantities for a given set of data will be observed to depend upon the value of towhich is used. The results of the calculations made by using various values of t o with a given set of T h and t data are presented in Table I. It should be also noted that with the schlieren optical system it is not thc desired quantity, dc/dr, which is obtained from the photographic plate; it is a numbeq proportiorial to it. Thus dc _ dr
-
dh
-%
+
(16)
Here, is a constant which incorporates dn/dc (the refractive index gradient), the bar angle a t which the photograph was taken, and the vertical magnification factor of the optical system. So, the quantity cBl (17) H. W. McCormick, J . Polymer Sci., 36, 341 (1959). (18) H. J. Cantow, Macromol. Chem., 30, 169 (1959).
THES E D I M E N T A T I O N VELOCITY EXPERIMEKT
165
method we have evolved. The results are tabulated in Table 11.
Table I : Comparison of Data for Quantities soeoand K with Several lo Values. Polystyrene S-103 at 59,780 r.p.m. with co = 0.313 g./lOO ml. hl e t t i od
LO
BaldwinfA
245 234 306
9/31,
"Best fit" a
soco (In
S)
4 114 4 92 5 08
K
0
-0 358 -0 347 -0 419
x
10'
2 6 2 9 1 8
Table 11: Data a t Four Rotational Speeds for the Systeni Polystyrene S-103 in Cyclohexane a t 34.2" rH
R. L. Baldwin, Btochem. J . , 65, 503 ( 1957). Soco
the concentration in the so-called plateau region, is given by the statement c,
=
le,, E:)(
dr
$ dr
= $
=
$A
(17)
-
to)
[1 - m
K u
x
10'
S0C0
to
where A is the area under the particular boundary gradient curve on the photographic plate. Thus, the essential working equation for the distribution of sedimentation coefficients becomes g*(sO) = r d ( t
lo
K
x
104
SnCO
to {(;)2
-
1 1 1 3 4
The asterisk has been used again because it was found to be necessary to make corrections for the spreading of the boundary due to diffusion. This required an extrapolatiori of the conventional kindl0I1' to give the function g(sO). The details are not provided here. The information from each experiment was read from the photographic plates by means of a Gaertner toolmaker microscope. From each photograph the location of the inner and outer index marks, the meniscus, arid from 19 to 23 values of dh/dr along the boundary gradient curve were determined. The readings always included data for the positions (dhldr),,, and (dhldr) = 0 at both ends of the curve. Calculations and Results 1 . IIse of rk, and r n in !he Determination of Sedimentation Coeflcients. In the evaluation of the sedimentation coefficient the position at the several times of the maximum of the refractive index gradient, r H , is often used in place of the boundary position as measured from the second moment, TI). I n order to study this choice further, sedimentation experiments a t four dif'ferent angular velocities with the system polystyrene 5-103 in cyclohexane at 34.2' were performed. The data were evaluated to give the quantities soco,to, K , and u, using thc two boundary positions. The time corrections, lo, were found by the
K
x
104
Soeo
to
K
x
104
Tb
Speed = 5!4,780 r.p.m. co = 0.313 g./lOO ml. 5.07 5.08 279 306 -0.36 -0 42 0 93 1.8 Speed = 56,100 r.p.rn. co = 0.64!1 g./100 ml. 4.76 4.88 188 246 -0 31 -0 44 1 04 1 0 Speed = 47,660 r.p.m. co = 0.7203 g./100 ml. 4.67 4.72 338 405 -0.19 -0.25 1.43 1.95 Speed = 39,460 r.p.m. ce = 0.7203 g./lO0 rnl. 4.77 4.78 58 1 642 -0.17 -0 214 0.599 0.522
Difference
0.2% 27 see. 15%
2 45% 58 sec. 29 5 %
1 .06% 67 sec. 26 o/c . .
0.2% 61 see. 20.594
Although the polymer is one of relatively narrow molecular weight distribution, the asymmetry of the boundary gradient curve introduces a significant error in the value of the parameter K if the displacement with time of the position r1I is used. Furthermore, the error in the determination of K so introduced does not seem to decrease with increased rotational speed. The spreading of the boundary gradient curve due to diffusion is reduced, but with the greater pressure at the cell bottom the actual distortion of the curve is enhanced. The difference in the socovalues is small and random, but the values of lo are difyererit in the two cases. In Fig. 1 and 2 are shown typical boundary gradient curves for the two polystyrenes. They were constructed from photographs taken a t late times during the progress of the experiments. Thc positions r f i and rl, are indicated. In the case of the morc nearly homogeneous sample (S-103) rli > rt,, hut with the highly polydispersc material (19F) these positions are reversed. The pressure dependencct of s tends to skew the curve away from the direction of sedimentation, since the leading molecules tend to he retarded ,
166
,JAMES E. BLAIRAND J.
w.W I L L L 4 M S
2. The Effect of Pressure on Sedimentation Velocity Behavior. During the course of any given sedimentation velocity experiment with a polymer-organic solvent system it appears that the effect of pressure on solute transport predominates over that of concentration dependence. These effects operate in opposite directions, as required by eq. 2a. The plot of In rt,/ro us. u2(t - to) would be linear in the ideal case, but it is concave downward when pressure dependence effects are present exclusively or are more important than those of concentration dependence. By reference to Fig. 3 it will be seen that this situation existed in the experiments we describe. 14
26
27
2Q 31 R , Gaertner units.
I
I
I
6
10
15
I
I
20
25
1
33
4
Direction of aedirnentation.
Figure 1. Boundary gradient curve for polystyrene S-103 in cyclohexane a t 34.2’; speed, 59,780 r.p.m.
0
- to) x 10-10. Plot of In rb/r0 us. w a ( t - t o )for polystyrene
30
oyt
Figure 3. S-103 in cyclohexane a t 34.2’; speed, 59,780 r.p.m. 2.0
From the values of K obtained from the experiments with polystyrene S-103, Table 11, the pressure dependence parameter W L was computed. From eq. 4 it follows that the quantity m/ro2should be a linear function of u2. The plot, open circles, is shown in Fig. 4.
d
.-
9
8
u
3
0
$ 1.0
40
\ .e
‘d
P’
s x
30
a
2 0
12
16
20
24
28
32
R, Gaertner units.
$ 2o g 3
4
Direction of sedimentation.
Figure 2. Boundary gradient curve for polystyrene 19F in cyclohexane a t 34.2”; speed, 59,780 r.p.m.
8
g 10 d u
2 0
0
in motion when they enter into regions of greater pressure. The “tail” on the weight distribution is in the direction of the heavier molecular weights. The Journal of Physical Chemistry
6
10
16 m/roa
x
20
25
108.
Figure 4. Plot of m/ro2 vs. square of rotational speed: 0, polystyrene 5-103; @, polyisobutylene F-22.
30
THESEDIMENTATION VELOCITY EXPERIMENT
167
Another point, indicated by a circle with cross, was obtained from the record of a n experiment with an isobutylene sample in the same solvent, cyclohexane, and a t the same temperature, 34.2'. From the location of this point one is led to suspect that the pressure dependence parameter m is determined primarily by the properties of the solvent. Another important consequence of the pressure dependence of s during a sedimentation velocity experiment is that the usual square radial dilution law is no longer valid; radial dilution now is described by eq. 11. This is to be seen in a plot of the quantity Arb2 vs. t where A , the area under the boundary gradient curve, is proportional to c,. The quantity Arb2 would be constant, i.e., independent of time, if there were no pressure dependence. I n Fig. 5 we present plots of Arb2 vs. time, using the data for two experiments at different speeds for the system polystyrene S-103 in cyclohexane. The broken lines represent the expected behavior in the absence of pressure dependence. Allowing for the pressure dependence of s, these plots would be expected to be straight lines of' positive slopes, since from eq. 11 it is seen that
[
Aoro2== Arb2 1 - m{(:)'
- l}]
or multiplied out
Arb2 == AoTo2
+ 2Soc,~2mAoTo2(t- t o )
(24)
Hence, a plot of Arb2 vs. time, as in Fig. 5, should be 8 straight line with slope 2 ~ ~ ~ , ~ ~ A ~ r ~ ~ m 3. Determination of the Molecular Weight Distribution. Data were taken from a single sedimentation velocity experiment a t 59,780 r.p.m. with a solution of polystyrene 19F in cyclohexane, concentration 0.504 g./100 ml., a t a temperature of 34.2'. From these data, the distribution of sedimentation coefficients, g(so) us. s, and the molecular weight distribution of the polymer, f ( M ) us. J4, were determined. I n processing the data, the value of the parameter K was taken as -0.45, as determined from Fig. 4. This number was felt to be more accurate than the value of K which had been computed from any single experiment.
40
(18) 44
As a first approximation, for small values of
Lj w-
c 42
4
40
1
+ 2SOC,W2(t- to+
-
F{($2- l}]
(19) 33
hence
2,000
(y1
=
L
2sOc,w2(t - to)/(l
+ msOc,w2(t - to))
(20)
I
8
4,000
6,000 8,000 t to, sec.
-
10,000
12,000 14,000
Figure 5. Plot of Arb2 vs. ( t - t o ) : upper curve, 56,100 r.p.m.; lower curve, 39,460 r.p,m.
Now, eq. 18 can be written in the form 2soc,w2m(t- to) 1 msO,w2(t - to)
+
or
For the values of the parameter applicable in these experiments, thle quantity 2soCp2(t- to) Is small, so eq. 22 can be approximated by
The transformation from sedimentation velocity distribution to the weight distribution was then made by means of eq. 14 in which were taken 1y = 0.50 and K = 1.47 X 1 0 F . The resultant f(A4) us. M curve is shown as the solid line of Fig. 6. The broken line in the figure indicates the weight distribution curve which is obtained when the pressure dependence of s is not taken into account. Numerical integrations of the corrected distribution curve of Fig. 6 were performed in order to obtain the average molecular weights &f,, and M,". By definition Volume 68, Number 1
January, 196k
JAMES E. BLAIRAND J. W. WILLIAMS
168
from Fig. 6 that failure to correct €or the effects of the pressure dependence of s results in a distribution which is richer in low weight polymers and poorer in the heavy species than is the actual case. and Table 111: Average Molecular Weight Data for Polystyrenes 19F and 5-103
The numerical results are M , 375,000.
=
207,000 and M , =
1QF
Sedimentation velocity with pressure correction (this work) Sedimentation velocity, no pressure correction (this work) Light scattering" Osmotic pressureb
2.40
2.00
1.60
M,
Mw
Mw/Mn
207,000
375,000
181
172,000
313,000 370,000
1.82 1.88
..... 197,000 =t5%
... .
S103
e.
2
Sedimentation equilibrium" 1.20
115,000
117,000 1.02
a D. J. Streeter and R. F. Boyer, Ind. Eng. Chem., 43, 1790 (1951). * L. D. Grandine, Jr., Dissertation, University of Wisconsin, 1952. c A. M. Linklater and J. W. Williams, unpublished.
E::
0.80
0.40
0
0
0.20
0.40
0.60 M
x
0.80
1.00
1.20
10-0.
Figure 6 . Molecular weight distribution of polystyrene 19F : , corrected for pressure dependence.
- - - - -, not corrected for pressure dependence;
I n Table I11 are summarized the values of Mn and M , and their ratios for the polystyrene 19F. Similar data for polystyrene S-103 are also included. It will be noted that the ratio M,/M,, is essentially unchanged when the pressure correction is neglected in determining the weight distribution of the polystyrene sample 19F, but both illwand M , are in error by a considerable amount. This confirms a similar observation which has been made by Billick.8 It is seen
The Journal of Physical Che mi8try
The important fact to be noted is that the distribution of molecular weights here obtained by velocity sedimentation is for all practical purposes that which corresponds to the distribution required by statistical analysis if the polymerization of the styrene terminates by the disproportionation mechanism. Exactly the same conclusion had been already obtained by using the sedimentation equilibrium experiment with another polystyrene of the same type,2a,2abut now advantage has been taken of the higher sensitivity of the transport process to heterogeneity. Acknowledgment. This research was made possible by generous grants from the U. S. Army Research Office (DA-ORD-11) and the University of Wisconsin Research Committee, using funds provided by the Wisconsin Alumni Research Foundation. Grateful acknowledgment is hereby made.
